1 preparing for the nj math assessments in the middle grades dr. eric milou rowan university...
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Preparing for the NJ Math Assessmentsin the Middle Grades
Dr. Eric Milou
Rowan University
Department of Mathematics
856-256-4500 x3876
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Overview
Conceptual vs. Procedural Debate– National Math Panel
Number Sense & Computation Proficiency
NJ mathematics assessments
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RhetoricNY Times (5/15/06)
In traditional math, children learn multiplication tables and specific techniques for calculating.
In constructivist math, the process by which students explore the question can be more important than getting the right answer, and the early use of calculators is welcomed.
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NCTM Focal Points (9/12/06)
September 12 Wall Street Journal article did not represent the substance or intent of the focal points.The focal points are not about the basics; they are about important foundational topics. NCTM has always supported learning the basics. Students should learn and be able to recall basic facts and become computationally fluent, but such knowledge and skills should be acquired with understanding.
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Education Week 11/1/06
We cannot afford to waste time on polarization. What is important is that we pragmatically address critical target areas to improve mathematics education. We cannot be distracted from our primary mission—to match tactical initiatives in other, newly technological societies that are snatching our competitive advantage in innovation—while we bicker over modest differences in approach. (Jere Confrey)
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Motivating Factors for Change
Society’s hate for mathematics that is prevalent and acceptable– 4 out of 10 adults hate mathematics* (twice as
many people said they hated math as said that about any other subject)
International test scoresIndustry concerns (no problem solving skills)National Council of Teachers of Mathematics (NCTM) Standards
*2005 AP-AOL News poll
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Compute the following:
4 x 9 x 25
900 - 201
50 ÷ 1/2
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What’s “Typical?” in US
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Third International Math & Science Study (TIMSS)
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8
16
52
18
46
20
48
19
37
59
00
10
20
30
40
50
60
70
80
Australia
Czech RepublicHong Kong
Japan
NetherlandsUnited States
Procedures vs. Concepts
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23.1
76.9
17
8378.1
21.9
0102030405060708090
100
Germany Japan United States
Stated Developed
Stated vs Developed
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Lesson Study
Demonstrates a procedure
Assigns similar problems to students as exercises
Homework assignment
Presents a problem without first demonstrating how to solve it
Individual or group problem solving
Compare and discuss multiple solution methods
Summary, exercises and homework assignment
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We need a BALANCE
Traditional text with conceptual supplement
Conceptual text (EM, CMP, Core-Plus) with computational supplement
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Conceptual Understanding
24 ÷ 4 = 6
24 ÷ 3 = 8
24 ÷ 2 =12
24 ÷ 1 = 24
24 ÷ 1/2 = ??
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Fractions - Conceptually
€
1
2+
1
3=
€
1
2+
1
3= More than 1 or Less
than 1
Explain your reasoning
€
3
6+
2
6=
5
6
€
2
5
The F word
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Which is larger?
2/3 + 3/4 + 4/5 + 5/6 OR 4
12.5 x 45 OR 4.5 x 125
1/3 + 2/4 + 2/4 + 5/11 OR 2
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Where’s the Point?
2.43 x 5.1 = 12393
4.85 x 4.954 = 240269
21.25 x 1.08 = 2295
1.25 x 64 = 80
4.688 x 1.355 = 635224
46.88 x 1.355 = 635224
4.688 x 135.5 = 635224
46.88 x 13.55 = 635224
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Computational Balance
1000 ÷ 1.49– Torture
Big Macs Sell for $1.49, how many Big Macs can I buy for $10.00?– 1 is $1.50– 2 are $3– 4 are $6– 6 are $9
Mental Mathematics
is a vital skill
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Computation is Important
Engaging & Active
Less passive worksheets
Creative!
More thinking & reasoning
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Name That Number - Computational Practice
Target #: 6
3 8 17
1 3
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Active Computation
Fifty (1, 2, 3, 4, 5, 6 and addition)Buzz (3)
Product Game
Wipe Out
Software: Math Arena
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Patterns
01
2
3
456
7
8
9
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Conceptual & Contextual
8+ 7 = ?
How do we teach this?xx x
x x xxx
xx
xx
xx
xxx
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17 - 8 =
1 7
- 8
//0 17
2 7
8 --> --> 10 --> --> --> --> --> --> --> 17
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1000 - 279 = ?
1000
−279
279 +1 = 280
+ 20 = 300
+700 = 1000
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Multiplication
13 x 17 = ?
1 3
x 1 7
1
2
9031
2 2 1
-------
-------
10 7
10
3
1 0 0
3 0
7 0
2 1
221
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Conceptual approach leads to ?
x 7
x
3
x2
3x
7x
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Algebra: (x + 3) (x + 7) =
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Contextual Problem Solving
Not more traditional word problems
Placing mathematical lessons into settings
Giving students a reason to learn the skill
Motivating students
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Example
You must select one spinner. Both spinners above will be spun once.
The spinner with the higher number showing wins $1,000,000 for that person.
Which spinner will you select?
4 6 8 5 9
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Spinner Example
BLUE ORANGE468468
555999
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Crossing the River
8 adults and 2 children need to cross a river and they have one small boat only available. The boat can hold ONLY:– One adult– One or two children
How many one-way trips does it take for all 8 adults and 2 children to cross?
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2006 NJ Assessment Data
NJASK36 non-calculator items (1/2 pt each)21 MC - calculator allowed - 1 pt each3 Open-ended - 3 pts each14 out of 33 points is a passing score
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2006 NJ Assessment Data
NJASK48 non-calculator items (1/2 pt each)24 MC - calculator allowed - 1 pt each5 Open-ended - 3 pts each17.5 out of 43 points is a passing score
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2006 NJASK 5, 6, 7
NJASK5 JPM was 18/39 (46%)
NJASK 6 JPM was 17/39 (44%)
NJASK 7 JPM was 13/39 (33%)
10 pts per cluster (one cluster with 9 pts)
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2006 NJ Assessment Data
GEPA
All items allow a calculator
30 Multiple choice items - 1 pt each
6 Open-ended - 3 pts each
25 out of 48 points is a passing score
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2006 NJ Assessment Data
HSPAAll items allow a calculator 30 Multiple choice items - 1 pt each6 Open-ended - 3 pts each20.5 out of 48 points is a passing score
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Assessments Points by Cluster
Cluster NJASK3 NJASK 4
NJASK 5
NJASK 6
Number 9 13 10 9
Geometry 8 10 9 10
Algebra 8 10 10 10
D/P/D 8 10 10 10
Total 33 43 39 39
“200” 14 17.5 18 17
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Assessments Points by Cluster
Cluster NJASK7 GEPA HSPA
Number 10 12 7
Geometry 9 12 12
Algebra 10 12 15
D/P/D 10 12 14
Total 39 48 48
“200” 13 25 20.5
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200 Score
Grade
3rd 14 33 42%4th 17.5 43 41%5th 18 39 46%6th 17 39 44%7th 13 39 33%8th 25 48 52%11th 20.5 48 43%
Just Proficient Means
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Implications & Inferences
NJ Assessments are rigorous and conceptualNJ Math Standards are well aligned with NJ assessmentsMost districts have a well aligned curriculum– Then, what’s wrong?
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Algebra Placement
Districts should not encourage all students to take Algebra I in grade 8; students should be taking Algebra I in grade 8 only if they are highly motivated, have a strong foundation in middle school mathematics, receive high grades in previous courses, intend to study calculus in high school, and only if the Algebra I courses are taught by teachers with mathematics certification.
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Fact #1
A
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Fact #2
B
43
Fact #3
C
44
Fact #4
D
45
Fact #5
E
46
Fact #6
F
47
Fact #7
G
48
Fact #8
H
49
Fact #9
I
50
What is this?
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What is this?
F A C E
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What If?
A B C
D E F
G H I
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Try Again
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Try Again
D E C A D E
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What’s the Point?
Isolated Facts– Less likely to retain information
Connected Facts, Patterns, Fact in Context– More likely to retain information
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Characteristics of a good mathematics program
CONCEPTUAL
CONTEXTUAL
CONSTUCTIVISM
COMPUTATION
TEST-PREP
